Definition of a Derivative: Interactive Quizzes and Practice

Question 1

What is the formal limit definition of the derivative of a function f(x) at a point x?

Accepted Answer

lim(h->0) [f(x+h) - f(x)] / h

Answer

lim(h->infinity) [f(x) - f(x+h)] / h

Answer

lim(h->infinity) [f(x+h) - f(x)] / h

Answer

lim(h->0) [f(x) - f(x+h)] / h

Question 2

Using the power rule, determine the derivative of the function f(x) = x^3.

Accepted Answer

3x^2

Answer

0

Answer

2x

Answer

x^2

Question 3

Determine the derivative of the polynomial function f(x) = x^4 - 3x^2 + 2x - 1.

Accepted Answer

4x^3 - 6x + 2

Answer

x^3 - 6x + 2

Answer

4x^3 + 6x - 2

Answer

4x^3 + 6x + 2

Question 4

Employing the chain rule, find the derivative of the exponential function f(x) = e^(3x).

Accepted Answer

3e^(3x)

Answer

e^(3x)

Answer

2e^(3x)

Question 5

Employing the sum/difference rule, find the derivative of the trigonometric function f(x) = cos(x) - sin(x).

Accepted Answer

-sin(x) - cos(x)

Answer

cos(x) - sin(x)

Answer

sin(x) + cos(x)

Answer

sin(x) - cos(x)

Question 6

Determine the derivative of the product function f(x) = x * e^(x).

Accepted Answer

e^(x) + x * e^(x)

Answer

x * e^(2x)

Answer

e^(x) - x * e^(x)

Answer

2x * e^(x)

Question 7

Employing both the product rule and chain rule, find the derivative of the function f(x) = sin(x^2).

Accepted Answer

2x * cos(x^2)

Answer

cos(x^2)

Answer

2x * sin(x^2)

Answer

x * cos(x^2)

Question 8

Which of the following is the limit definition of the derivative of a function f(x)?

Accepted Answer

lim_(h->0) (f(x + h) - f(x)) / h

Answer

lim_(h->0) (f'(x + h) - f'(x)) / h

Answer

lim_(x->h) (f(x + h) - f(x)) / h

Question 9

What is the derivative of the function f(x) = x^2 ?

Accepted Answer

2x

Answer

x

Answer

1

Answer

x^3

Question 10

What is the derivative of the function f(x) = x^3 * e^x?

Accepted Answer

(3x^2 + x^3) * e^x

Answer

x^3 * e^(x^2)

Answer

(3x^2 + e^x) * x^3

Answer

(3x^2 * e^x) + x^3

Question 11

Which of the following is the derivative of the function f(x) = ln(x)?

Accepted Answer

1/x

Answer

ln(x^2)

Answer

x

Question 12

What is the derivative of the function f(x) = sqrt(x)?

Accepted Answer

1 / (2 * sqrt(x))

Answer

2 * sqrt(x)

Answer

1 / sqrt(x)

Answer

1 / (2 * x)

Question 13

Which of the following is the derivative of the function f(x) = 1 / x?

Accepted Answer

-1 / x^2

Answer

x / 2

Answer

1 / x^2

Answer

-x / 2

Question 14

Which of the following is the derivative of the function f(x) = e^(x^2)?

Accepted Answer

2xe^(x^2)

Answer

-2xe^(x^2)

Answer

1 / (2xe^(x^2))

Answer

x^2e^(x^2)

Question 15

What is the mathematical definition of the derivative of a function f(x)?

Accepted Answer

limₗ∂→0 [f(x + h) - f(x)] / h

Answer

[f(x + h) + f(x)] / h

Answer

∫[f(x + h) - f(x)] / h dx

Answer

limₗ∂→∞ [f(x + h) - f(x)] / h

Question 16

Determine the derivative of the function f(x) = sin(x²) by applying the chain rule.

Accepted Answer

2xcos(x²)

Answer

xcos(x²)

Answer

cos(x²) / 2x

Answer

sin(x)cos(x²)

Question 17

Which of the following functions has a derivative equal to zero at all values of x?

Accepted Answer

f(x) = c, where c is any constant

Answer

f(x) = sin(x)

Answer

f(x) = e¹ˣ

Answer

f(x) = x²

Question 18

Which of the following functions is not differentiable at x = 0?

Accepted Answer

f(x) = |x|

Answer

f(x) = e¹ˣ

Answer

f(x) = sin(x)

Answer

f(x) = x²

Question 19

Which of the following is the limit definition of the derivative of a function **f(x)**?

Accepted Answer

lim(h->0) [f(x + h) - f(x)] / h

Answer

lim(h->0) [f'(x) - f(x)] / h

Answer

lim(x->a) f(x) / x

Answer

lim(x->a) [f(x) - f'(x)] / (x - a)

Question 20

What does the derivative of a function represent?

Accepted Answer

The instantaneous rate of change

Answer

The area under the curve

Answer

The maximum rate of change

Answer

The average rate of change

Question 21

Which rule is used to find the derivative of a power function, such as **x^n**?

Accepted Answer

Power rule

Answer

Chain rule

Answer

Product rule

Answer

Quotient rule

Question 22

Find the derivative of the function **f(x) = x^3 + 2x^2 - 5**.

Accepted Answer

3x^2 + 4x

Answer

3x^2 + 2

Answer

x^3 + 4x

Answer

3x^2 - 4x

Question 23

Which rule is used to find the derivative of a product of two functions, such as **f(x)g(x)**?

Accepted Answer

Product rule

Answer

Quotient rule

Answer

Chain rule

Answer

Power rule

Question 24

Which rule is used to find the derivative of a composition of functions, such as **f(g(x))**?

Accepted Answer

Chain rule

Answer

Quotient rule

Answer

Power rule

Answer

Product rule

Question 25

What is the formula for the chain rule?

Accepted Answer

f'(g(x))g'(x)

Answer

f'(x) + g'(x)

Answer

f'(x) - g'(x)

Answer

g'(x) - f'(x)

Question 26

Find the derivative of the function **f(x) = sin(x^2 + 1)**.

Accepted Answer

2x*cos(x^2 + 1)

Answer

cos(x^2 + 1)

Answer

sin(x^2 + 1)

Answer

-2x*cos(x^2 + 1)

Question 27

Find the derivative of the function f(x) = x^3 + 2x^2 - 5x + 1 using the limit definition of the derivative.

Accepted Answer

f'(x) = lim (h->0) [(f(x+h) - f(x))/h] = lim (h->0) [(x^3 + 3x^2*h + 3xh^2 + h^3 + 2x^2 + 4xh + 2h^2 - 5x - 5h + 1 -x^3 - 2x^2 -x - 1)/h]

Answer

f'(x) = 3x^2 - 4x - 5

Answer

f'(x) = x^2 + 2x - 5

Answer

f'(x) = 3x^2 + 4x - 5

Question 28

Calculate the derivative of the function y = (x^2 + 1)(x - 2) using the product rule.

Accepted Answer

y' = (x^2 + 1)'(x - 2) + (x^2 + 1)(x - 2)' = (2x)(x - 2) + (x^2 + 1)'

Answer

y' = x^2 - 4x + 2

Answer

y' = 2x + 1

Answer

y' = (2x)(x - 2)

Question 29

Find the derivative of the function h(x) = e^(x^2).

Accepted Answer

h'(x) = e^(x^2) * (x^2)'

Answer

h'(x) = e^x

Answer

h'(x) = x^2e^(x^2)

Question 30

Which of the following functions has a derivative that is always zero?

Accepted Answer

f(x) = c, where c is a constant

Answer

f(x) = x^2

Answer

f(x) = sin(x)

Answer

f(x) = e^x

Question 31

Find the equation of the tangent to the curve y = f(x) at the point where x = 2, given that f'(2) = 3.

Accepted Answer

y - f(2) = f'(2) * (x - 2)

Answer

y = 3x - 4

Answer

y = f(2)

Answer

y - f(2) = -3(x - 2)

Question 32

Find the derivative of the function y = ln(x^2 + 1).

Accepted Answer

y' = (1/(x^2 + 1)) * (x^2 + 1)'

Answer

y' = 1/x

Answer

y' = 2x

Answer

y' = 1/(x^2 + 1)

Question 33

Which of the following is a real-world application of derivatives?

Accepted Answer

Calculating the slope of a hill for road construction.

Answer

Determining the volume of a cylinder.

Answer

Finding the area of a circle.

Question 34

Which of the following correctly defines the limit definition of the derivative of a function f(x) at a point x=a?

Accepted Answer

lim_(h->0) [(f(a+h)-f(a))/h]

Answer

lim_(x->a) [f(x+h)-f(x)]

Answer

lim_(h->0) [f(a-h)-f(a)]

Answer

lim_(x->0) [f(x)-f(a)]

Question 35

Use the power rule to calculate the derivative of the function f(x) = x^3 - 2x^2 + 5.

Accepted Answer

3x^2 - 4x

Answer

x^2 + 4x

Answer

x^2 - 4x

Answer

-3x^2 - 4x

Question 36

Find the derivative of f(x) = sin(2x) by utilizing the chain rule.

Accepted Answer

2cos(2x)

Answer

cos(2x)

Answer

sin(2x)

Answer

-2cos(2x)

Question 37

Using the limit definition, determine the derivative of the function f(x) = sqrt(x).

Accepted Answer

1/(2sqrt(x))

Answer

2/sqrt(x)

Answer

sqrt(x)

Answer

1/sqrt(x)

Question 38

Find the derivative of f(x) = |x| at the point x= 0.

Accepted Answer

Does not exist

Answer

-1

Answer

0

Question 39

Identify the function that possesses a constant derivative.

Accepted Answer

f(x) = 5

Answer

f(x) = e^x

Answer

f(x) = x^2

Answer

f(x) = sin(x)

Question 40

What is the derivative of the function f(x) = sin(x)?

Accepted Answer

cos(x)

Answer

-cos(x)

Answer

sin(x)

Answer

1/cos(x)

Question 41

Using the product rule, find the derivative of the function f(x) = (x² + 1)(x - 2).

Accepted Answer

3x² - 4x + 1

Answer

(x² + 1) + (x - 2) * 2x

Answer

2x² + 1 - 2x² + 4x

Answer

(x² + 1) * (x - 2) * 2

Question 42

Apply the quotient rule to find the derivative of the function f(x) = (x³ - 2) / (x² + 1).

Accepted Answer

(x⁴ + 3x² + 2) / (x² + 1)²

Answer

((x² + 1) * 3x² - (x³ - 2) * 2x) / (x² + 1)²

Answer

(x³ - 2) / (x² + 1)²

Answer

(x² + 1) * 3x² - (x³ - 2) * 2 / (x² + 1)

Question 43

Using the chain rule, find the derivative of the function f(x) = sin(2x + 1).

Accepted Answer

2cos(2x + 1)

Answer

1/cos(2x + 1)

Answer

2sin(2x + 1)

Answer

cos(2x + 1)

Question 44

What is the derivative of the function f(x) = |x|?

Accepted Answer

1 if x > 0, -1 if x < 0, undefined if x = 0

Answer

x / |x|

Answer

|x|

Answer

-|x|

Question 45

Find the equation of the tangent line to the curve y = x³ - 2x² + 1 at the point (1, 0).

Accepted Answer

y = x - 1

Answer

y = x² - 2x

Answer

y = x³ - 2x²

Answer

y = 0

Question 46

Which statement about derivatives is NOT true?

Accepted Answer

Derivatives are always positive.

Answer

Derivatives can be used to find critical points.

Answer

Derivatives measure the rate of change.

Answer

Derivatives can be used to determine concavity.

Question 47

Which of the following techniques is NOT used to find the derivative of a function?

Accepted Answer

Integration by parts

Answer

Limit definition

Answer

Power rule

Answer

Chain rule

Question 48

What is the derivative of the function f(x) = 3x² + 2x - 1?

Accepted Answer

6x + 2

Answer

9x + 2

Answer

6x² + 2x

Answer

3x + 2

Question 49

The derivative of a function f(x) at a point x = a represents:

Accepted Answer

The instantaneous rate of change of f(x) at x = a

Answer

The average rate of change of f(x) over an interval containing a

Answer

The slope of the tangent line to the graph of f(x) at x = a

Question 50

What is the limit definition of the derivative of a function *f(x)*?

Accepted Answer

f'(x) = lim_(h->0) [f(x + h) - f(x)] / h

Answer

f'(x) = lim_(h->0) [f(x) - f(h)] / h

Answer

f'(x) = lim_(x->a) [f(x) - f(a)] / (x - a)

Answer

f'(x) = lim_(x->0) [f(x + h) - f(x)] / h

Question 51

What is the derivative of the function *f(x) = x² sin(x)*?

Accepted Answer

2x sin(x) + x² cos(x)

Answer

2x sin(x) - x² cos(x)

Answer

2x cos(x)

Answer

x² cos(x)

Question 52

What is the derivative of the function *f(x) = (x³ + 2x)²*?

Accepted Answer

2(x³ + 2x)(3x² + 2)

Answer

2x(x³ + 2x)

Answer

(3x² + 2)²

Answer

6x² + 4x

Question 53

What is the derivative of the function *f(x) = 1/x*?

Accepted Answer

-1/x²

Answer

1/x²

Answer

x

Answer

-x

Question 54

What is the derivative of a constant function?

Accepted Answer

Zero

Answer

The constant itself

Answer

One

Answer

Undefined

Question 55

What is the equation of the tangent line to the curve *y = x² + 2x* at the point *(1, 3)*?

Accepted Answer

y = 4x - 1

Answer

y = 3x

Answer

y = x + 2

Answer

y = 2x + 1

Question 56

What is the derivative of the function *f(x) = e^x*?

Accepted Answer

e^x

Answer

x * e^x

Answer

1/e^x

Answer

e^(x-1)

Question 57

If the velocity of an object is given by *v(t) = 2t + 1*, what is the object's acceleration?

Accepted Answer

2

Answer

4t + 1

Answer

2t

Answer

1